Numerical solution of PDE:s, Part 5: Schrödinger equation in imaginary time

In the last post, I mentioned that the solution of the time dependent 1D Schrödinger equation

can be written by expanding the initial state in the basis of the solutions of time-independent Schrödinger equation

and multiplying each term in the expansion by a complex-valued time dependent phase factor:

Now, assuming that the energies are all positive and are ordered so that is the smallest of them, we get an interesting result by replacing the time variable with an imaginary time variable . The function is then

which is a sum of exponentially decaying terms, of which the first one, decays slowest. So, in the limit of large , the wave function is

i.e. after normalizing it, it’s approximately the same as the ground state . Also, if is a large number, we have

or

So, here we have a way to use the TDSE to numerically calculate the ground state wavefunction and corresponding energy eigenvalue for any potential energy function . This is very useful, as the ground state can’t usually be solved analytically, except for some very simple potential energy functions such as the harmonic oscillator potential.

To test this imaginary time method, I will approximately calculate the ground state of an anharmonic oscillator, described by a Schrödinger equation

Here the shape of the anharmonic potential has been plotted to the same images.

The problem with this method for finding a ground state is that if the system has more degrees of freedom than a single coordinate x, the matrices in the linear systems needed in the time stepping quickly become very large and the calculation becomes prohibitively slow. To make this worse, the Crank-Nicolson method of time stepping can’t be parallelized for multiple processors. However, there is another way to compute the evolution of a wave function in imaginary time, which is called Diffusion Monte Carlo, and that is easily parallelizable. DMC is one practical way for calculating ground state wave functions for multi-particle systems such as a helium or a lithium atom.